2.10: Summary

Last updated
Save as PDF

Page ID: 8317

\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \) \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)\(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\) \(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\)\(\newcommand{\AA}{\unicode[.8,0]{x212B}}\)

In some materials a property will be the same, irrespective of the direction in which it is measured, but this is not always the case. On completion of this TLP, you should now understand the concept of anisotropy, and be able to appreciate that a response can be non-parallel to the applied stimulus. Anisotropy in a range of properties has been discussed, including electrical and thermal conductivity, diffusion, dielectric permittivity, and optical properties. You should also now be familiar with the use of representation surfaces for a range of anisotropic properties, including the basis behind their mathematical description.

Anisotropic properties are exploited in many applications. In polarised-light microscopy, a quartz wedge can be used to determine birefringence and optical sign. Liquid crystals have electronic uses such as displays, and the liquid crystalline state has advantages in the processing of polymers (such as Kevlar). The anisotropic thermal conductivity in polymer thin films has use in microelectronic devices, for example, solid-state transducers.

Anisotropic properties described by higher than second rank tensors (not discussed here) can also have useful applications. Examples include:

Piezoelectricity (relating an applied stress to the induced polarisation)
The electro-optic effect (when a field causes a change in the dielectric impermeability)
Elastic compliance and elastic stiffness (relating stress and strain)
Piezo-optical effect (when a stress induces a change in refractive index)
Electrostriction (strain arising from an electric field)

Non-tensor properties can also demonstrate anisotropy; for example, yield stress can vary with direction of applied stress.

Going further

Books

A. Putnis, Introduction to Mineral Sciences, CUP, 1992 (specifically Chapter 2, "Anisotropy and physical properties").
R.E. Newnham, Structure-Property Relations Relations (Crystal chemistry of non-metallic materials), Springer-Verlag, 1975
D.R. Lovett, Tensor Properties of Crystals, IOP, 1999
P.J. Collings and M. Hird, Introduction to Liquid Crystals: Chemistry and Physics, Taylor & Francis, 1997

More advanced and detailed books:

C. Kittel, Introduction to Solid State Physics, 7th Edition 1995
J.F. Nye, Physical Properties of Crystals: Their Representation by Tensors and Matrices, Oxford, 2nd Edition 1985
E. Hecht, Optics, Addison-Wesley, 4th Edition 2001

Websites

Polymers and Liquid Crystals
An award-winnning website based at Case Western Reserve University in the USA, with a Virtual Textbook and Virtual Laboratory.
Introduction to Photoelasticity
A TLP covering many features of birefringence under polarised light.
Theoretical methods for Mineral Sciences research
Summary of phase transitions and formation of domains in perovskites.
Optical Birefringence
This comprehensive introduction to optical birefringence is part of the excellent award-winning Molecular Expressions website based at Florida State University in the USA.
Birefringence in calcite crystals
Also part of the Molecular Expressions website, this is an interactive Java tutorial.

For some light relief, take a look at:

The Optical Indicatrix - And Why I Love It So Much